Derived H-module endomorphism rings

Abstract

Let H be a Hopf algebra, A/B be an H-Galois extension. Let D(A) and D(B) be the derived categories of right A-modules and of right B-modules, respectively. An object M-. is an element of D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings E-A(M-.) =. circle plus i is an element of zHom(D(A))(M-. , M-. [i]) and E-B(M-.) = circle plus i is an element of z Hom(D(B))(M-., M-. [i]). If H is a finite-dimensional semi-simple Hopf algebra, then E-A(M-.) is a graded sub-algebra of E-B(M-.). In particular, if M is a usual A-module, a necessary and sufficient condition for E-B(M) to be an H*-Galois graded extension of E-A(M) is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.